Target Space Duality as a Symmetry of String Field Theory

Toroidal backgrounds for bosonic strings are used to understand target space duality as a symmetry of string field theory and to study explicitly issues in background independence. Our starting point is the notion that the string field coordinates $X(\sigma)$ and the momenta $P(\sigma)$ are background independent objects whose field algebra is always the same; backgrounds correspond to inequivalent representations of this algebra. We propose classical string field solutions relating any two toroidal backgrounds and discuss the space where these solutions are defined. String field theories formulated around dual backgrounds are shown to be related by a homogeneous field redefinition, and are therefore equivalent, if and only if their string field coupling constants are identical. Using this discrete equivalence of backgrounds and the classical solutions we find discrete symmetry transformations of the string field leaving the string action invariant. These symmetries, which are spontaneously broken for generic backgrounds, are shown to generate the full group of duality symmetries, and in general are seen to arise from the string field gauge group.